Question

Consider the two parallel coaxial disks of diameters \(a\) and \(b\), shown in Fig. P13-131. For this geometry, the view factor from the smaller disk to the larger disk can be calculated from $$ F_{i j}=0.5\left(\frac{B}{A}\right)^{2}\left\\{C-\left[C^{2}-4\left(\frac{A}{B}\right)^{2}\right]^{0.5}\right\\} $$ where, \(A=a / 2 L, B=b / 2 L\), and \(C=1+\left[\left(1+A^{2}\right) / B^{2}\right]\). The diameter, emissivity and temperature are \(20 \mathrm{~cm}, 0.60\), and \(600^{\circ} \mathrm{C}\), respectively, for disk \(a\), and \(40 \mathrm{~cm}, 0.80\) and \(200^{\circ} \mathrm{C}\) for disk \(b\). The distance between the two disks is \(L=10 \mathrm{~cm}\). (a) Calculate \(F_{a b}\) and \(F_{b a}\). (b) Calculate the net rate of radiation heat exchange between disks \(a\) and \(b\) in steady operation. (c) Suppose another (infinitely) large disk \(c\), of negligible thickness and \(\varepsilon=0.7\), is inserted between disks \(a\) and \(b\) such that it is parallel and equidistant to both disks. Calculate the net rate of radiation heat exchange between disks \(a\) and \(c\) and disks \(c\) and \(b\) in steady operation.

Short answer

Question: Calculate the view factors F_ab and F_ba, the net rate of radiation heat exchange between disk a and disk b, and the net rate of radiation heat exchange between disk a, a new disk c, and disk b when a new disk is inserted between them with given emissivity. Given: Diameter a = 20 cm Diameter b = 40 cm Distance L = 10 cm Emissivity of disk a = 0.8 Emissivity of disk b = 0.6 Emissivity of disk c = 0.9 Temperature of disk a, T_a = 500 K Temperature of disk b, T_b = 300 K

Step-by-step solution

Calculate A, B, and C

We are given the values of diameter \(a\), \(b\), and distance \(L\). We can calculate \(A\), \(B\) and \(C\) using the given expressions. $$A= \frac{a}{2L}, B= \frac{b}{2L}, C=1+\frac{1+A^2}{B^2}$$ Using the given values, \(a=20cm\), \(b=40cm\), and \(L=10cm\), we can calculate \(A\), \(B\), and \(C\).

Calculate the view factors F_ab and F_ba

Now that we have calculated \(A\), \(B\), and \(C\), we can use them to calculate the view factors \(F_{ab}\) and \(F_{ba}\) using the given formula for the view factor. $$F_{ab} = 0.5\left(\frac{B}{A}\right)^{2}\left\\{C-\left[C^{2}-4\left(\frac{A}{B}\right)^{2}\right]^{0.5}\right\\}$$ Since the disks are coaxial, the view factors are related by reciprocity. $$F_{ba} = \frac{A_2}{A_1}F_{ab}$$ Where \(A_1\) and \(A_2\) are the areas of disks \(a\) and \(b\), respectively.

Calculate the net rate of radiation heat exchange between disk a and disk b

Now that we have the view factors between the disks, we can calculate the net rate of radiation heat exchange between them using the following equation. $$Q_{net} = F_{ab}A_1\sigma[T_1^4 - T_2^4]$$ where \(\sigma\) is the Stefan-Boltzmann constant, and \(T_1\) and \(T_2\) are the absolute temperatures of disks \(a\) and \(b\).

Calculate the net rate of radiation heat exchange between disk a, disk c, and disk b with disk c inserted between them

Since disk c is inserted between disks a and b, we can calculate the net rate of radiation heat exchange between disks a and c, and disks c and b using the following equation for the net rate of radiation exchange. $$Q_{net,ac} = F_{ac}A_{a}\sigma[T_{a}^4 - T_{c}^4]$$ $$Q_{net,cb} = F_{cb}A_{c}\sigma[T_{c}^4 - T_{b}^4]$$ Since the problem statement mentions that disk c has negligible thickness and is equidistant from both disks a and b, we can assume \(F_{ac} = F_{ab}\) and \(F_{cb} = F_{ba}\) (as found in Step 2) in steady operation. Finally, we can calculate the net rates of radiation heat exchange between disks a and c, and disks c and b. After following these steps, we will have solved the given exercise by calculating the view factors and the net rates of radiation heat exchange between the various pairs of disks.

Key concepts

View Factor Calculation

The view factor, also known as the configuration factor or shape factor, is a crucial component in understanding radiative heat transfer between surfaces. It represents the fraction of the radiation that leaves one surface and directly reaches another. In essence, it quantifies how well two surfaces 'see' each other. Calculating the view factor requires a deep understanding of the geometry involved.

For situations involving simple shapes such as parallel plates, concentric cylinders, or coaxial discs (as in our exercise), specific formulas like the one provided can be applied. Accurate calculation of view factors is essential since they directly affect the heat exchange estimations. For instance, in the problem at hand, the step-by-step solution demonstrates the use of provided geometrical relationships to calculate the view factors between two disks. A meticulous approach ensures the precise estimation of how much thermal radiation is exchanged between the disk surfaces.

Stefan-Boltzmann Law

The Stefan-Boltzmann law is fundamental to understanding thermal radiation. It states that the total energy radiated per unit surface area of a black body in unit time is directly proportional to the fourth power of the black body's absolute temperature. Mathematically, it is expressed as: \[E = \sigma T^4\] where \(E\) is the emissive power, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} W/m^2K^4\)), and \(T\) is the absolute temperature in kelvins.

The law is applied in calculating the radiation heat transfer from one surface to another by factoring in the temperatures of the surfaces and the Stefan-Boltzmann constant. In our exercise, it empowers us to quantify the amount of heat exchanged between the two disks, which is crucial for understanding the radiative heat exchange process.

Radiative Heat Transfer

Radiative heat transfer is a mode of heat transfer that occurs through electromagnetic waves, predominantly in the infrared spectrum. It differs from conduction and convection as it does not require a medium; it can occur through a vacuum.

In the context of our exercise, radiative heat transfer is between two disks at different temperatures and emissivities. The net rate of radiant heat exchange is governed by factors such as the view factors of the surfaces, their emissivities, and their temperatures. This physical phenomenon is modeled mathematically to predict the behaviour of systems in a range of engineering applications, such as in HVAC design, spacecraft thermal control, and even in electronic devices.

Thermal Radiation

Thermal radiation refers to the emission of electromagnetic waves from the surface of an object due to the object's temperature. All objects with a temperature above absolute zero emit thermal radiation. The quality and quantity of this radiation depend on properties such as the object's temperature and emissivity.

Emissivity is a measure of how effectively a surface emits thermal radiation in comparison to a black body at the same temperature. The problem under consideration involves calculating the radiation heat transfer between objects at different temperatures, which means understanding thermal radiation principles is necessary. Applying thermal radiation concepts, we can determine how the presence of the third disc (disc c) affects the thermal radiation exchange in the system.